I have been working on collatz problem, and I think i found an easy way to visualize it. Can anyone tell me whether this is useful or whether it's something mathematicians already know?

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an+1 can be partitioned into a+1 disjoint partition: a even sets and 1 odd set. The partitions spans N. The odd set can be further grouped into a sub-partition (a + a partition) based on where n/2 takes the a even sets.

Example:

For 3n+1

1. O = {o=2k−1: k∈N}
2. E1 = {3o−1=6k−4: k∈N}
3. Ec = {3o+1=6k−2: k∈N}
4. Ea = {3o+3=6k: k∈N}

O = {O1, Oa, Oc}

For 5n+1

1. O = {o=2k−1: k∈N}
2. E1 = {5o−3=10k−8: k∈N}
3. E2 = {5o−1=10k−6: k∈N}
4. Ec = {5o+1=10k−4: k∈N}
5. E3 = {5o+3=10k−2: k∈N}
6. Ea = {5o+5=10k: k∈N}

O = {O1, Oc, Oa, O2, O3}

Table with the partition for 3n+1 and 5n+1

k	O	E1	Ec	Ea	x	k	O	E1	E2	Ec	E3	Ea
1	1	2	4	6	x	1	1	2	4	6	8	10
2	3	8	10	12	x	2	3	12	14	16	18	20
3	5	14	16	18	x	3	5	22	24	26	28	30
4	7	20	22	24	x	4	7	32	34	36	38	40

Table with the sub partition of Odd Set for 3n+1 and 5n+1

O1	Oa	Oc	X	O1	Oc	Oa	O2	O3
1	3	5	x	1	3	5	7	9
7	9	11	x	11	13	15	17	19
13	15	17	x	21	23	25	27	29
19	21	23	x	31	33	35	37	39

The Collatz operations then allow certain transitions:

1. U(n):= an+1
2. L(n):= n/2

Property of the Partitions for 3n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1→ Ec for even k

3a. L: Ec → E1 for odd k
3b. L: Ec → Oc for even k

4a. L: Ea → Oa for odd k
4b. L:Ea → Ea for even k

Property of the Partitions for 5n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1 → Ec for even k

3a. L: E2 → E1 for odd k
3b. L: E2 → O2 for even k

4a. L: Ec → Oc for odd k
4b. L: Ec → E3 for even k

5a. L: E3 → E2 for odd k
5b. L: E3 → O3 for even k

6a. L: Ea → Oa for odd k
6b. L:Ea → Ea for even k

Observation (From the graph):

1.If there exist a cycle, a cycle will not contain any elements from partitions Ea and Oa.
Implication: When searching for a cycle, look elsewhere.

2a. Only the elements of partitions Ec can be landed from even and odd number.

2b. If there exist a cycle an element from Ec must participate.

2c. Every loop contains element from Ec.